You should not have problems.
My recent experience in Greece (mainly in Crete, off-season):
Athens airport: no problem, English is understood, signs are also in English, personnel speak English. There is no problem in customs (like most countries: it is not necessary to know local language).
Car rental: no problem: you will have documents in English, and people will speak at minimum basic English (with accent)
Taxi: no problem, but better to have the address written down (Latin characters are fine)
Restaurants: In a small one, they will find a way to communicate with you, usually using a son/daughter, or asking an local guest to help. They also have tourist menus (with images) or they show you what they can provide.
Tourist attraction: signs, entries, prices are displayed also in English (Euro currency), most people know enough English.
Most directions on streets are also in English.
You may want to learn Greek alphabet, but not necessary. You may not be able to read all signs (e.g. pharmacy, but logo will help).
You may lose some local chatting (because you do not know Greek, and locals will often not go much further than basic English), but this may not be a problem.
You may try also with Italian, German, French, or Spanish, especially with older people.
Note: I recommend Google Translator (with offline dictionary) on phone, just if you are curious about meaning of some words, and on restaurants ask for a WiFi: the English name of Greek specialties may not say much for us, but Google Image Search helps.